According to Jewish law, if a Jewish man gives a single Jewish woman an item of intrinsic value with both clearly understanding intent of marriage, in front of two ritual witnesses, then they are married.
A rabbi has no intrinsic power to "marry" a couple. He's there to referee that they're following the rules (e.g. make sure he's actually giving her a ring that's his, and not borrowing one from a friend); and as it's a societal mess for 14-year-olds to go running off with their high school crushes, to enforce some sort of social norms.
Please don't use the term "shiksa" by the way, it's offensive. You can simply say "a non-Jewish woman."
Jewish law recognizes no marriage whatsoever between a Jew and a non-Jew. You can march them down the aisle in front of a million rabbis and he can give her the Hope Diamond, but nothing happens.
As far as US law is concerned, there are nuances between states but broadly the general rule is if two people believe that this person is "clergy" and this clergyperson believes they've been married, s/he can sign their marriage license. (In some states, clergy have to register or show credentials with the state first. In others you can pretty much dub someone clergy on your own. In Wyoming I think you can officiate your own wedding.) Thus if a Hindu man and Jewish woman meet and they don't care about intermarriage, if they get a Reform rabbi (the Reform movement allows intermarriage), Hindu pandit, Universalist Unitarian minister, or any other flavor of clergy they want to sign their certificate, the state will take it. (The state may ask to see the Reform rabbi's ordination certificate from Hebrew Union College to prove that s/he is a rabbi, but will otherwise keep itself out of religious matters to ask any more questions about the couple.)